How to rig a plate so that it points to a target, with off-centre tilt, and constraints on elongation and tilt
Disclaimer: This solution is not quite geometrically perfect, it doesn't take into account the disk radius and angle when adjusting the height to the target, so the disk doesn't necessarily touch the target at close ranges, but otherwise it seems to work, and ended up being relatively simple.

Just setup a single armature with two bones, so that the middle hinge is at half the whole height of the armature. In this case the armature is 1 unit high, so the tail of the second bone is at $0.5$ units in Z.

Create your $Target$ empty exactly at the tip of the second bone.

Add a single Transform constraint to the whole armature, and set it up to transform the Z location of the source object into Scale transform of the armature, and pick the $Target$ object as source.



Getting the right values was sort trial and error, but since the armature is 1 unit high, I suspect the min and max values for both source Z coordinate are respectively $0$ and $1$. Set the max scale to whatever height you want to limit the arms to; set space from World to Local.

Now enter Edit Mode in your armature, select the tip bone and add a bone constraint of the Damped Track type and point it to the same $Target$ empty. Track with Y and set Z as Up. You may also want to add a Limit Rotation to bound it to the desired values.



And is about it for the setup. Just create your disk object and parent it to the tip bone (not the whole armature). Make sure to add a Limit Scale constraint with all axis set to $1.0$ so it doesn't change size with target height.

Moving the $Target$ empty around should give you the effect you want while always bending at the middle hinge point.